Markus–Yamabe Conjecture for Asymptotic Stability of Planar Piecewise Linear Systems

We present in this paper a detailed study on the Markus–Yamabe conjecture in planar piecewise linear systems. We consider discontinuous piecewise linear systems with two zones separated by a straight line, in which every subsystem is asymptotically stable. We prove the existence of limit cycles under explicit parameter conditions and give more different counterexamples to the Markus-Yamabe conjecture in addition to the counterexamples given by Llibre and Menezes. In particular, we consider continuous planar piecewise linear systems. For such a system with n + 1 zones separated by n parallel straight lines in phase space, we prove that if each of subsystems is asymptotically stable, then this system has a globally asymptotically stable equilibrium point, therefore the Markus–Yamabe conjecture still holds. Some examples are given to illustrate the main results.Mathematics Subject Classification (2020) 34C05 · 34C07 · 37G15

MATHEMATICAL MODELING OF THE MECHANICAL CHARACTERISTICS OF ELECTRIC MOTORS WITH A PHASE ROTOR UNDER STABLE MOTION OF MECHANISMS OF LIFTING AND TRANSPORTATION MACHINES

A method of mathematical synthesis of the equations of the mechanical characteristics of an electric motor with a phase rotor, corresponding to various resistance steps in the rotor circuit and approximated by straight lines, when simulating the unsteady movement of the mechanisms of hoisting-and-transport machines, is presented.

EFFICIENT LINE DETECTION METHOD BASED ON 2D CONVOLUTION FILTER

The article proposes an efficient line detection method using a 2D convolution filter. The proposed method was compared with the Hough transform, the most popular method of straight lines detection. The developed method is suitable for local detection of straight lines with a slope from -45˚ to 45˚. Also, it can be used for curve detection which shape is approximated with the short straight sections. The new method is characterized by a constant computational cost regardless of the number of set pixels. The convolution is performed using the logical conjunction and sum operations. Moreover, design of the developed filter and the method of filtration allows for parallelization. Due to constant computation cost, the new method is suitable for implementation in the hardware structure of real-time image processing systems.

Nanoimprinted and Anodized Templates for Large-Scale and Low-Cost Nanopatterning

Nanopatterning to fabricate advanced nanostructured materials is a widely employed technology in a broad spectrum of applications going from spintronics and nanoelectronics to nanophotonics. This work reports on an easy route for nanopatterning making use of ordered porous templates with geometries ranging from straight lines to square, triangular or rhombohedral lattices, to be employed for the designed growth of sputtered materials with engineered properties. The procedure is based on large-scale nanoimprinting using patterned low-cost commercial disks, as 1-D grating stamps, followed by a single electrochemical process that allows one to obtain 1-D ordered porous anodic templates. Multiple imprinting steps at different angles enable more complex 2-D patterned templates. Subsequently, sputtering facilitates the growth of ferromagnetic antidot thin films (e.g., from 20 to 100 nm Co thick layers) with designed symmetries. This technique constitutes a non-expensive method for massive mold production and pattern generation avoiding standard lithographical techniques. In addition, it overcomes current challenges of the two-stage electrochemical porous anodic alumina templates: (i) allowing the patterning of large areas with high ordering and/or complex antidot geometries, and (ii) being less-time consuming.

Graphically interpreting how incision thresholds influence topographic and scaling properties of modeled landscapes

We examine the influence of incision thresholds on topographic and scaling properties of landscapes that follow a landscape evolution model (LEM) with terms for stream-power incision, linear diffusion, and uniform uplift. Our analysis uses three main tools. First, we examine the graphical behavior of theoretical relationships between curvature and the steepness index (which depends on drainage area and slope). These relationships plot as straight lines for the case of steady-state landscapes that follow the LEM. These lines have slopes and intercepts that provide estimates of landscape characteristic scales. Such lines can be viewed as counterparts of slope–area relationships, which follow power laws in detachment-limited landscapes but not in landscapes with diffusion. We illustrate the response of these curvature–steepness index lines to changes in the values of parameters. Second, we define a Péclet number that quantifies the competition between incision and diffusion, while taking the incision threshold into account. We examine how this Péclet number captures the influence of the incision threshold on the degree of landscape dissection. Third, we characterize the influence of the incision threshold using a ratio between it and the steepness index. This ratio is a dimensionless number in the case of the LEM that we use and reflects the fraction by which the incision rate is reduced due to the incision threshold; in this way, it quantifies the relative influence of the incision threshold across a landscape. These three tools can be used together to graphically illustrate how topography and process competition respond to incision thresholds.

VYSHKOVO REFORMED CHURCH – THE INCREDIBLY VALUABLE ARCHITECTURAL MONUMENT OF WESTERN UKRAINE

The church, built according to various sources in the IX-XIII centuries, is interesting for its monumental, restrained architecture and preserved interior. The clear straight lines of the building with sharp elements remind us of the Gothic style, although according to legend it was built much earlier. Covered with drawings of the XVIII century. the ceiling of the temple is related to the famous painted ceiling of the church in Chetfalva. Next to the church is a beautiful, shingled wooden bell tower of the Marmara type. The monument is located in a picturesque, well-groomed courtyard, surrounded on all sides by a wall, where exotic trees and flowers grow. In front of the entrance gate there are several Hungarian carved wooden sculptures erected in memory of important historical events in Vyshkovo. The purpose of the article is to attract the attention and acquaint readers with the incredibly valuable architectural monument of Western Ukraine – Vyshkovo Reformed Church. The article presents the preconditions and the process of construction, determining the type of materials used, design analysis, detailed consideration of decorative elements and various plans of the oldest church in Transcarpathia. My own research of this architectural object, which I have been conducting for three months, describes to us: the methods of construction at that time, the execution of the frames of door and window openings, and the decoration of the facade of the church. Religion has always played an important role in the life of the people of Vyshkovo. People are proud of their church, they constantly visit it. Being in the middle of the building, there is a feeling of authenticity and complete immersion in the events described by the priest, because of the very good acoustics of this ancient church.

A Full-Period Mathematical Model for a Hybrid-Rotor Bearingless Switched Reluctance Motor

A bearingless switched reluctance motor (BSRM) has the combined characteristics of a switched reluctance motor (SRM) and a magnetic bearing. The hybrid-rotor BSRM (HBSRM) discussed in the paper has a twelve-pole stator and an eight-pole hybrid rotor, which is composed of a cylindrical rotor and a salient-pole rotor. Although the asymmetry of the hybrid rotor makes the structure and magnetic field of the HBSRM more complex, it can always produce a significant amount of magnetic pulling force to levitate a rotor shaft at all the rotor angular positions of each phase, which is not available in a traditional BSRM. The classical mathematical model for a conventional BSRM is valid only when its rotor rotates from the start of the overlap position to the aligned position, and the radial force and torque derived from this model are discontinuous at the aligned positon, which is harmful to the motor’s stable operation. In this paper, a full-period mathematical model on the assumption that the gap permeance is cut apart by straight lines or improved elliptical lines for a 12/8-pole HBSRM is provided. On the basis of this mathematical model, the continuity of the radial force and torque at all the rotor angular positions can be guaranteed, and the fine characteristics of this mathematical model have been verified by simulations.

A new geometry definition and generation method for a face gear meshed with a spur pinion

The conventional tooth surface of a face gear is difficult to manufacture, and the cutter for the face gear cutting is not uniform even though the parameters of the pinion mating with the face gear slightly change. Based on the analysis of the geometry features of the tooth surface, a new developable ruled surface is defined as the tooth flank of the face gear, for which the most important geometry feature is that the flank could be represented by a family of straight lines, hence it could be generated by a straight-edged cutter. The mathematical models of the new ruled tooth surface, the cutter and the generation method are presented, the deviation between the ruled surface and the conventional surface, the correction of the ruled surface to reduce the deviation are investigated through numerical examples. The manufacturing process is simulated by VERICUT software, and the results demonstrate that even when the principle deviation is added to the machined deviation, the absolute deviation is on the micro-scale. The meshing and contact simulation shows that the new surface could obtain good meshing performance when the number of face gear teeth is greater than three times the number of pinion teeth. This research provides a new method for manufacturing face gears.

Plains in a Special Case. The Relationship Between the Planes

Annotation: To develop students' understanding of straight lines and planes and to develop skills and competencies in working on related issues. The listener must complete the given task on A4 paper with the necessary tools. Keywords: straight lines, perpendicular, horizontal projection, frontal projections.